Oct
18: Ekman currents
Nansen’s qualitative
arguments:
Figure 9.2 The balance of forces acting on an iceberg in a wind on a
rotating Earth.
Fridtjof
Nansen noticed that wind tended to blow ice at an
angle of 20°-40° to the right of the wind in the
Nansen
argued that three forces must be important:
Wind Stress, W;
Friction F (otherwise the
iceberg would move as fast as the wind);
Coriolis
Force, C.
Nansen
argued further that the forces must have the following attributes:
Drag must be opposite the direction of the ice's velocity;
Coriolis
force must be perpendicular to the velocity;
The forces must balance for steady flow.
W + F + C = 0
Nansen’s
ideas led to the work of Walfrid Ekman.
The Ekman balance is another simple balance between two forces.
Steady surface wind
stress, when balanced by the Coriolis force, sets up
the so-called Ekman transport in the upper ocean
wind-driven mixed layer.
The depth range
over which this dynamical balance can be achieved can be estimated by a simple
scale analysis:
If we have only Coriolis forces balancing the frictional mixing of wind momentum input
by the wind then we have the Ekman equations:
Consider the order
of magnitude of the terms:
fv is o(fV)
Avd2u/dz2 is o(AV/de2)
where de is some vertical boundary
layer scale over which the momentum from the wind is mixes into the ocean by
vertical turbulent eddies.
The ratio of these
two is the Ekman number:
Ek = AV/ de
2 V/f = A/f de 2
A typical eddy
viscosity would be 10-2 m2s-1 and f is about 10-4
s-1
For an Ek of O(1) we need de
2 = A/f or de
= 
The depth range
over which the Ekman balance can apply is very
limited, of order only tens of meters below the sea surface.
The Ekman equations can be solved exactly for the case of Av = constant. The solution
is a spiraling velocity pattern that decays with depth.
For the case of a
wind stress directed in the positive y-direction the solution is:
where 
is termed the Ekman depth
As z becomes negative, the magnitude of the
velocity decays exponentially and the direction rotates clockwise (for f>0).
This is pattern of
currents is termed the “Ekman spiral.”
Figure 9.3. Ekman current generated by a 10 m s-1 wind at 35°N (from Stewart)
At z=0
This is a surface
current 45o (i.e. 
) to the right of the wind.
We can verify that
the surface wind stress condition is met by evaluating
which gives the maximum
surface velocity in terms of the wind stress:
For a wind stress
of 0.1 Pa and an assumed Av
of 10-2 m2s-1 that gave rise to a o(10 m) Ekman depth, we get
Uo
~ (0.1)(10-3 )(10-2.10-4)-1/2
= 0.1 m s-1 or 10 cm s-1
Since a stress of
0.1 Pa is produced by a roughly 10 m/s wind, this calculation suggests surface
wind driven Ekman currents are typically order(100)
times smaller than the wind speed.
This result also
shows that the same wind produces a different maximum surface current at
different latitudes.
From the equation
for the idealized Ekman spiral solution, we see that
velocity pattern depends on the details of the eddy viscosity profile.
In reality, a
distinct Ekman velocity spiral is seldom observed in
the ocean.
Important concepts
·
Ekman number o(1) implies the direct
influence of the winds is limited to a relatively shallow depth in the ocean
·
a fundamental depth scale arises which shows how the depth of the Ekman
layer scales with latitude and magnitude of the mixing coefficient (the
wind-driven currents decay roughly exponentially with this scale)
·
the velocity pattern is predominantly to the right (left) of the
wind in the north (south) hemisphere
Progressive vector diagram, using daily averaged currents
relative to the flow at 48 m, at a subset of depths from a moored ADCP at
37.1°N, 127.6°W in the California Current, deployed as part of the Eastern Boundary Currents experiment. Daily
averaged wind vectors are plotted at midnight UT along the 8-m relative to 48-m
displacement curve. Wind velocity scale is shown at bottom left. (From: Chereskin, T. K., 1995: Evidence for an Ekman
balance in the California Current. J. Geophys. Res., 100, 12727-12748.)